https://awgutierrez.github.io/tags/normed-spaces/Normed SpacesHugoBlox Kit (https://hugoblox.com)en-usTue, 07 Apr 2026 00:00:00 +0000https://awgutierrez.github.io/media/icon_hu_eaf62b3751e6012a.pnghttps://awgutierrez.github.io/tags/normed-spaces/https://awgutierrez.github.io/publications/d_weak_convergence/Tue, 07 Apr 2026 00:00:00 +0000https://awgutierrez.github.io/publications/d_weak_convergence/<h2 id="main-results">Main results</h2>https://awgutierrez.github.io/publications/lp_boundary/Fri, 19 Oct 2018 00:00:00 +0000https://awgutierrez.github.io/publications/lp_boundary/<h2 id="main-results">Main results</h2> <div class="callout flex px-4 py-3 mb-6 rounded-md border-l-4 bg-green-100 dark:bg-green-900 border-green-500" data-callout="success" data-callout-metadata=""> <span class="callout-icon pr-3 pt-1 text-green-600 dark:text-green-300"> <svg height="24" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" d="M9 12.75L11.25 15L15 9.75M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0"/></svg> </span> <div class="callout-content dark:text-neutral-300"> <div class="callout-title font-semibold mb-1">Metric functionals on \(\ell_1^n\)</div> <div class="callout-body"><p>Every metric functional on \(\ell_1^n\) is either internal or of the form </p> $$ h(x) = \sum_{i\in I}\epsilon_i x_i + \sum_{i\not\in I}(|x_i - \mu_i| - |\mu_i|),$$<p> where \(\empty \neq I \subseteq [n]\), \((\epsilon_i) \in \{-1, 1\}^I\), and \((\mu_i) \in \mathbb{R}^{[n]\setminus I}\).</p></div> </div> </div> <div class="callout flex px-4 py-3 mb-6 rounded-md border-l-4 bg-green-100 dark:bg-green-900 border-green-500" data-callout="success" data-callout-metadata=""> <span class="callout-icon pr-3 pt-1 text-green-600 dark:text-green-300"> <svg height="24" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" d="M9 12.75L11.25 15L15 9.75M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0"/></svg> </span> <div class="callout-content dark:text-neutral-300"> <div class="callout-title font-semibold mb-1">Metric functionals on \(\ell_p^n\) with \(1 < p < \infty\)</div> <div class="callout-body"><p>Every metric functional on \(\ell_p^n\) with \(1 < p < \infty\) is either internal or of the form </p> $$ h(x) = - \sum_{i\in [n]} \mu_i x_i\,,$$<p> where \((\mu_i)\in \mathbb{R}^n\) with \(||\mu||_{q} = 1\) and \(q :=p(p-1)^{-1}\).</p></div> </div> </div>